Breaking Rotational Symmetry in Supertwisted WS2 Spirals via Moiré Magnification of Intrinsic Heterostrain

Twisted stacking of van der Waals materials with moiré superlattices offers a new way to tailor their physical properties via engineering of the crystal symmetry. Unlike well-studied twisted bilayers, little is known about the overall symmetry and symmetry-driven physical properties of continuously supertwisted multilayer structures. Here, using polarization-resolved second harmonic generation (SHG) microscopy, we report threefold (C3) rotational symmetry breaking in supertwisted WS2 spirals grown on non-Euclidean surfaces, contrasting the intact symmetry of individual monolayers. This symmetry breaking is attributed to a geometrical magnifying effect in which small relative strain between adjacent twisted layers (heterostrain), verified by Raman spectroscopy and multiphysics simulations, generates significant distortion in the moiré pattern. Density-functional theory calculations can explain the C3 symmetry breaking and unusual SHG response by the interlayer wave function coupling. These findings thus pave the way for further developments in the so-called “3D twistronics”.


Methods.
Synthesis of WS2 nanostructures. The WS2 spiral structures were synthesized following previous reports. 1, 2 WO3 nanoparticles (Sigma Aldrich, nanopowder, <100 nm particle size) dispersed in ethanol were drop cast onto a 300 nm SiO2/Si substrate and dried prior to the reaction. 100 mg WS2 precursor powder (Alfa Aesar) in an alumina boat was placed in the first zone of a three-zone furnace. The substrates were placed between the second and the third zone with the polished side facing up. 1 g CaSO4‧2H2O powder (Sigma Aldrich) was placed upstream from the heating zone as the water vapor source and heated with heating tapes. With 100 sccm argon flowing at 800 torr, the second zone was heated to 1200 °C at a rate of 20 °C/min, and simultaneously the third zone was heated to 700 °C at the same rate, while the CaSO4‧2H2O was not heated. Once the furnace temperatures were reached, the CaSO4‧2H2O was heated to 90 °C to release the water vapor. After all the temperatures were stabilized, the WS2 boat was pushed into the second zone by a magnet coupled positioner and a quartz rod to initiate the reaction. After 15 min reaction, the furnace was opened and cooled down rapidly.

Polarization resolved second harmonic generation (SHG) intensity patterns and imaging.
SHG measurement was performed using a Ti:Sapphire femtosecond laser with ~800 nm of wavelength, ~100fs of pulse width, and 80 MHz of repetition rate. The laser beam was mechanically chopped for signal demodulation to suppress the background noise. The polarization of the input laser beam was controlled by a half-wave plate and focused through a 50X long working distance objective lens. The generated SHG signal at ~400 nm was collected through the same objective lens and separated from the input fundamental laser beam by a harmonic separator. A polarizer filtered the polarization of the SHG signal, and color filters were used to block the fundamental laser beam. A photomultiplier tube (PMT) collected the SHG signal, which was demodulated at the modulation of the frequency of the input laser beam by a lock-in amplifier.
Density-functional theory (DFT) calculations. DFT calculations were performed using Vienna ab initio simulation package with projector-augmented wave pseudopotentials. 3,4 The generalized gradient approximation of Perdew−Burke−Ernzerhof was used for the exchangecorrelation functionals. 5 The cutoff energy for the plane-wave basis was set to 450 eV. The convergence criteria of energy and forces were set to 1×10 −4 eV and 0.05 eV Å −1 , respectively. The van der Waals interaction was induced by using the D3 correction scheme of Grimme. 6 For charge density difference calculations in twisted bilayer WS2, we explored 4×4 supercell as discussed in previous studies. 7,8 The bottom pristine monolayer WS2 with D3h symmetry has zigzag crystal orientation aligned with x-axis; the top layer with ~3% strain is twisted clockwise by ~7º with respect to the bottom layer. The 4×4 grid of unstrained WS2 primitive cell was combined with the corresponding size strained WS2 to build the supercell of homostructure and allow relaxation for DFT calculations. The k-point sampling was obtained from the Monkhorst−Pack scheme with a 4×4×1 mesh.
Raman and AFM characterization. Raman spectra were measured by a Raman spectrometer (Renishaw Inc.) with a 488 nm laser as the excitation source. Sample morphology was characterized by an AFM system (Vistascope, Molecular Vista) in the tapping mode.   Figure  1. The blue arrow indicates the average "armchair" orientations (δ) at the position of "c".
When considering the material absorption, the SH field from the Nth layer can be written as: 10 where Γ = Δ ( −1) − ( −1) /2 defines the attenuation factor of the SH field from the Nth layer due to the material absorption, ϕ is the azimuthal angle between the armchair direction and the incident laser polarization direction, α means absorption coefficient of WS2 to be 0.086 nm -1 at 400 nm, 11 t is the thickness between the surface and the Nth layer of the sample, δN is the twisting angle of the Nth layer relative to the layer "c" denoted in the left of Figure   S2a, Δ = 4 0 ( ) (− (2 ) − ( )) is the wave-vector difference between the fundamental (ω) and the SH (2ω) light, λ0 is the wavelength of the fundamental laser to be 800 nm, n(2ω) and n(ω) mean the refractive index of WS2 to be 3.78 at 400 nm and 4.10 at 800 nm, 12 respectively. According to the SH field superposition theory, the total SH field ( ⃗ 2 ) is the vector additions of dipole moments from each electrically decoupled individual layer ( ⃗ 2 ): Thus, the total SHG ( 2 ∝ | ⃗ 2 | 2 ) at different positions were calculated in Figure S2b. These patterns show full six-fold symmetry, indicating that the traditional SH field superposition theory cannot explain the experimental two-lobe SHG patterns in the supertwisted spiral (Figure 1). The average "armchair" orientations of δ fitted by SH field  We determined the existence of heterostrain between adjacent layers in the supertwisted spiral structure using steady-state finite-element method (FEM) calculations (COMOSOL multiphysics with the Solid Mechanics module). Due to the challenge of building a supertwisted geometry above the curved surface, the regular spiral model was created in Figure S4a. An out-of-plane force was applied beneath the center of the aligned spiral to qualitatively mimic the influence of protrusion on the formation of the supertwisted spiral in Figure S4b, based on the mechanism of the "non-Euclidean" twist in our previous publication. 2 We note that as a result of the spiral shape in the three-dimensional space, its parallel cross section is not a complete triangle but a concave quadrilateral shape, as shown in Figure 4b. When applying the out-of-plane force beneath the spiral, the deformation increases from the upper layer to the lower layer (Figure 4b), suggesting the presence of heterostrain in the supertwisted spiral.
We note that the FEM simulations only qualitatively demonstrated the presence of distinct tensile strain among neighboring layers in the supertwisted structure, because real strain values in each layer are largely defined by the size of the protrusion and the relaxation process during the sample synthesis, which is challenging to be accurately simulated by COMSOL. However, real values of heterostrains in the supertwisted spirals do not affect the main conclusion of our work because a small heterostrain can be magnified about 10-fold, thus significantly distorting the moiré pattern and breaking the overall symmetry, as shown in Figure 4d. |, 15,16 where the Poisson ratio ( ) is 0.22. 14 The twist angle (θ) is tailored to + ± along two lattice vectors by heterostrain in Figure S5d. Note S1. Eliminating the impact of oblique incidence on the SHG response.
The polarization resolved SHG patterns in Figure 1 were measured at positions away from the protrusion beneath the supertwisted spirals to ensure the normal incidence. Furthermore, due to the large ratio between the width (~200 nm) and the height (~0.7 nm) of the step, the stair angle of the spiral is too small to affect the SHG patterns. It has been reported that when the incident angle of the fundamental laser is η, the SHG intensity can be expressed as ∝ | 2 • 3 | 2 . 17,18 Thus, in our spiral structure, the η value is so small (<< 1º) that the influence of oblique incidence on SHG is negligible.
Note S2. Modified bond additivity model to fit SHG patterns of the supertwisted spiral.
The polarization dependent SHG signal can be evaluated by the bond additivity mode (BAM). 19 Here, we consider a phenomenological mode by adding an effective bond to introduce the observed asymmetric two-fold signal in the SHG pattern.
The twist in the spiral structure will not affect the D3h symmetry of each WS2 layer, so the 6fold symmetry of the SHG pattern was initially expected to be preserved. However, our SHG signal exhibits a stronger angular preference closer to the center of the supertwisted spiral, where the heterostrain between neighboring twisted layers is enhanced. Based on the bond additivity model, the asymmetric phenomena can be explained by introducing an effective bond to describe the moiré magnification of heterostrain and the resultant elongated hexagon moiré patterns in Figures 5d and S5.
We assume the incident polarized light with the electric field as = 0̂= 0 (sin , cos , 0) where is the angle between incident laser polarization and the y-axis of the lab coordinate, as shown in Figure 5a. For the WS2 layer with D3h symmetry, the SHG comes from three effective linear bonds. Each of these bonds has the only nonzero hyperpolarizability tensor element (2) = 0 , where indicates the direction along this bond. Moreover, we consider an additional effective bond with an angle relative to -axis of the lab coordinate, and its (2) = 0 Θ . Now, in the lab coordinate, we have the total hyperpolarizability tensor where ̂= (sin , cos , 0) is the direction of the additional bond in the lab coordinate, stands for the different WS2 bonds within the D3h symmetry bond, and indicate the bond coordinate. We note that the component of the effective bonds will not affect the SHG result.
The SHG intensity can be calculated as where is the number of atoms participating in the SH response. Close to the center region, the easy deformation nature and the overall C3 rotational symmetry breaking response to the second harmonic measurement.